#374202

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
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

1468 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:

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

Polycyclic presentation:

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

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, 281398);