#6811672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2265 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)(7,10)(8,12)(9,11)(13,14)(15,22)(16,26)(17,24)(18,20)(19,23)(21,25)(27,28);(2,3)(5,7,10,6)(8,12,9,11)(14,20)(15,16)(23,24)(25,27);(1,2,4,3)(13,15,18,16)(14,21,20,28)(17,25,19,27)(22,24,26,23);(5,8,10,9)(6,11,7,12)(13,16,18,15)(14,22,20,26)(17,25,19,27)(21,24,28,23);(1,3,4,2)(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,23,20,24)(15,25,16,27)(21,22,28,26)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257386, #72257507, #72264140
maximal normal subgroup of	#72257386, #72257507, #72264140
maximal quotient of	#72265663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718868);