#406344

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1302 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)(7,9)(8,10)(11,12)(13,14,17,22,19,24,18,23)(15,25,28,36,29,35,27,20)(16,21,31,38,32,39,33,40)(26,37,30,43,42,44,41,34);(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,30)(17,27)(18,28)(19,29)(21,37)(22,35)(23,25)(24,36)(26,31)(32,41)(33,42)(34,38)(39,44)(40,43);(1,2,3,4)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,38)(23,40)(24,39)(25,37)(27,41)(28,30)(29,42)(35,44)(36,43);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,38,39,40)(26,41,42,30)(34,44,43,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313540);