#6491829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308062, #72308384
maximal normal subgroup of	#72308062, #72308384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6399025);