#393340

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	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2362 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)(7,11)(8,9)(10,12)(13,14)(15,18,22,25)(16,19)(17,20)(21,24,27,28)(23,26);(6,9)(10,11)(13,15)(14,18,20,25)(16,21)(17,22)(19,24,26,28)(23,27);(5,7,8,12)(6,10,9,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(2,4)(16,23)(19,26)(21,27)(24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,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, 300536);