#122437

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	#1459
prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,11,14,26)(4,8)(5,18,20,9)(6,10)(7,13)(12,27,15,29)(16,31,32,24)(17,25)(19,22)(21,28)(23,30)(33,34,38,44,39,45,52,59,40,46,36,42,37,43,53,60)(35,47,50,63,49,58,64,56,51,61,48,62,54,41,55,57);(1,3,6,14)(2,7,10,22)(4,15,17,12)(5,19,20,13)(8,23,25,21)(9,26,18,11)(16,28,32,30)(24,27,31,29)(33,35)(34,41)(36,50)(37,54)(38,48)(39,49)(40,51)(42,57)(43,61)(44,56)(45,47)(46,58)(52,55)(53,64)(59,62)(60,63);(1,4,6,17)(2,8,10,25)(3,12,14,15)(5,16,20,32)(7,21,22,23)(9,24,18,31)(11,27,26,29)(13,28,19,30)(33,36,39,53,40,38,37,52)(34,42,45,60,46,44,43,59)(35,48,49,55,51,50,54,64)(41,56,47,62,58,57,61,63)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=dhi,e^b=efhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29633);