#377043

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1854 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)(9,10,12,11)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,27,26,23);(1,2,4,3)(5,6)(7,8)(14,18)(19,26)(20,25)(21,28);(6,8)(13,15,17,23,16,22,24,27)(14,19,20,21,18,26,25,28);(1,3,4,2)(9,11,12,10)(14,18)(19,26)(20,25)(21,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=eh,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, 284239);