#360026

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1464 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:

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

Polycyclic presentation:

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

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, 267222);