#188340

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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 ▷ C₂² ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=di,d^c=di,g^c=gh

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, 95536);