#21484

order	256
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	Aut(G) = C₂²×#72218857 (order 6144) · Inn(G) = C₂×#120 · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#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 ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 653 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#21484⋊C₃, #72236787, #72236797, …
maximal subgroup of	#495754, #495756, #495779, …
maximal normal subgroup of	#495754, #495756, #495779, …
maximal quotient of	#208808, #208809, #208810, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14472);