#354845

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1778 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262041);