#6814492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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
prime radical	#42927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721688);