#405302

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307850, #72307868, #72307882
maximal normal subgroup of	#72307850, #72307868, #72307882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312498);