#125545

order	512
nilpotency class	4
isoclinism	isoclinic class of #125508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fgi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fi,g^a=gi,c^b=chi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi

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, 32741);