#363118

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1492 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=efi,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi

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, 270314);