#361787

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1902 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268983);