#6683342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2089 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)(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,17,25,20)(16,26,27,21)(18,22,24,28);(5,7,8,12)(6,9,10,11)(13,15,19,25)(14,17,23,20)(16,21,27,26)(18,22,24,28);(1,2,3,4)(13,16,19,27)(14,21,23,26)(15,24,25,18)(17,22,20,28);(6,10)(9,11)(13,17,19,20)(14,15,23,25)(16,22,27,28)(18,21,24,26);(7,12)(9,11)(13,18)(14,22)(15,16)(17,26)(19,24)(20,21)(23,28)(25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308624, #72308819
maximal normal subgroup of	#72308624, #72308819

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590538);