#375742

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=efh,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282938);