#6445074

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25123	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281225, #72282293, #72308528
maximal normal subgroup of	#72281225, #72282293, #72308528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352270);