#417769

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#449	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315860, #72315897, #72316213
maximal normal subgroup of	#72315860, #72315897, #72316213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324965);