#21483

order	256
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) = C₂²×#72218857 (order 6144) · Inn(G) = #120 · Out(G) = C₂³×S₄
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#120
commutator	C₂³	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	C₄×#120
prime radical	#3025	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 185 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,11,7,5,10,19)(3,8,15,22,14,13,21,28)(4,9,18,24,17,16,23,29)(12,25,27,32,26,20,30,31);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,20)(10,21)(11,22)(16,25)(17,26)(18,27)(19,28)(23,30)(24,31)(29,32);(1,4)(2,9)(3,12)(5,16)(6,18)(7,17)(8,20)(10,23)(11,24)(13,25)(14,26)(15,27)(19,29)(21,30)(22,31)(28,32);(1,5,6,19,7,2,10,11)(3,13,15,28,14,8,21,22)(4,16,18,29,17,9,23,24)(12,25,27,32,26,20,30,31)

Polycyclic presentation:

2^8|a^2=g,d^2=g,g^2=h,b^a=be,c^a=cf,f^b=fh,e^c=eh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#21483⋊C₃, #72236846, #72236849, …
maximal subgroup of	#136829, #136830, #495677, …
maximal normal subgroup of	#136829, #136830, #495677, …
maximal quotient of	C₈×B(4,𝔽₂), C₈×#458, #208806, …
automorphism group of	#1400

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14471);