#393345

order	512
nilpotency class	3
isoclinism	isoclinic class of #195963
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1594 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)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,21,18,20,25,15)(16,19,24,27,23,26,28,22);(6,9)(10,11)(13,15,17,20,18,21,25,14)(16,22,24,26,23,27,28,19);(5,7,8,12)(6,10,9,11)(13,16,18,23)(14,19,20,26)(15,22,21,27)(17,24,25,28);(2,4)(16,23)(19,26)(22,27)(24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi,d^c=di

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, 300541);