#358223

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265419);