#6810510

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42433	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717706);