#6810589

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
prime radical	#25736	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 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)(13,14,19,17)(15,20,18,26)(16,23,25,28)(21,27,24,22);(2,3)(5,7,8,12)(6,9,10,11)(13,15,19,18)(14,20,17,26)(16,22,25,27)(21,28,24,23);(1,2,4,3)(7,12)(9,11)(13,16,19,25)(14,21,17,24)(15,22,18,27)(20,28,26,23);(6,10)(9,11)(13,17,19,14)(15,20,18,26)(16,24,25,21)(22,28,27,23);(1,3,4,2)(14,17)(20,26)(21,24)(23,28)

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,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717785);