#6810557

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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
prime radical	#25734	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2083 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)(7,11)(8,10)(9,12)(13,14,19,16)(15,22,25,28)(17,21,18,26)(20,27,23,24);(2,3)(5,7,8,12)(6,9,10,11)(13,14,19,16)(15,23,25,20)(17,26,18,21)(22,24,28,27);(1,2,4,3)(7,12)(9,11)(13,15)(14,20)(16,23)(17,24)(18,27)(19,25)(21,28)(22,26);(6,10)(9,11)(13,16,19,14)(15,23,25,20)(17,21,18,26)(22,24,28,27);(1,3,4,2)(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717753);