#221390

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
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	C₂×#2456	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1580 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)(15,19,25,28,23,26,24,27);(1,2)(3,4)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,28)(22,26);(5,7,8,12)(6,9,10,11)(15,23)(19,26)(24,25)(27,28);(2,4)(13,16,18,17)(14,20,22,21)(15,24,23,25)(19,27,26,28)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,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, 128586);