#512204

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33171	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 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)(5,6)(7,10)(8,9)(11,12)(14,20)(15,26)(16,21)(17,28)(18,22)(23,24)(25,27);(6,9)(10,12)(13,14,21,15,18,24,28,27,19,25,17,23,22,26,16,20);(5,7,8,11)(6,10,9,12)(13,15,16,14,22,20,17,26,19,23,28,25,18,27,21,24);(2,4)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319432, #72319515, #72319524
maximal normal subgroup of	#72319432, #72319515, #72319524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419400);