#23794

order	256
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	Aut(G) = C₂²×#72224760 (order 6144) · Inn(G) = #127 · Out(G) = C₂³×S₄
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2397
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 125 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#23794⋊C₃
maximal subgroup of	#150326, #150327, #150344, …
maximal normal subgroup of	#150326, #150327, #150344, …
maximal quotient of	#244113, #244114, #244115, …
automorphism group of	#1398

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16782);