#6814458

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2090 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)(13,14,18,21)(15,19,24,25)(16,17,26,20)(22,23,27,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,23)(17,27)(18,24)(20,22)(21,25)(26,28);(1,2,3,4)(13,16,14,20,18,26,21,17)(15,22,25,23,24,27,19,28);(6,10)(9,11)(13,14,18,21)(15,19,24,25)(16,20,26,17)(22,28,27,23);(7,12)(9,11)(13,17,21,26,18,20,14,16)(15,23,19,22,24,28,25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721654);