#6814509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#39960	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1933 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:

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

Polycyclic presentation:

2^9|b^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,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, 6721705);