#6459739

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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	#72216151
maximal normal subgroup of	#72216151
maximal quotient of	#72216058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366935);