#6545017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#38079	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 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)(13,14,17,22,18,21,27,28)(15,19,23,26,20,16,24,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,19,21,16)(17,23,27,24)(22,26,28,25);(6,9)(10,12)(13,16,18,19)(14,20,21,15)(17,26,27,25)(22,23,28,24);(5,7)(6,10)(8,11)(9,12)(14,21)(15,20)(22,28)(23,24);(7,11)(10,12)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452213);