#6465910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 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)(13,14,19,22)(15,20,25,28)(16,17,21,18)(23,24,26,27);(2,3)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,24)(19,25)(21,23)(22,28);(1,2,4,3)(13,16,18,14,19,21,17,22)(15,23,24,28,25,26,27,20);(1,3,4,2)(13,17,19,18)(14,21,22,16)(15,24,25,27)(20,26,28,23);(5,7,8,11)(6,10,9,12)(15,25)(20,28)(23,26)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285009, #72285079
maximal normal subgroup of	#72285009, #72285079

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373106);