#379383

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,25,29,37)(16,30,33,34)(17,20,18,23)(21,26,35,28)(22,32,38,31)(27,36,42,44)(39,41,43,40);(5,7,8,12)(6,9,10,11)(14,20,24,23)(15,26,29,28)(21,35)(22,34,38,30)(25,37)(27,40,42,41)(36,44)(39,43);(13,15,18,28,19,29,17,26)(14,21,20,25,24,35,23,37)(16,27,32,41,33,42,31,40)(22,36,34,39,38,44,30,43);(1,3,4,2)(6,10)(9,11)(13,16,17,31,19,33,18,32)(14,22,23,30,24,38,20,34)(15,27,26,40,29,42,28,41)(21,36,37,43,35,44,25,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286579);