#412322

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1646 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)(10,12)(13,14,18,26,17,20,16,21)(15,22,25,19,24,28,23,27);(5,6)(7,8)(13,15,17,24)(14,19,20,27)(16,25,18,23)(21,28,26,22);(2,4)(6,8)(14,20)(19,27)(21,26)(22,28);(9,10,11,12)(14,20)(19,27)(21,26)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264567, #72264576, #72264579, …
maximal normal subgroup of	#72264567, #72264576, #72264579, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319518);