#6577085

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282430, #72282710, #72282899
maximal normal subgroup of	#72282430, #72282710, #72282899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484281);