#365657

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272853);