#359937

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267133);