#370325

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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)(7,9)(8,10)(11,12)(13,14,19,22,18,23,17,33)(15,20,27,35,26,36,25,42)(16,28,32,38,30,44,31,21)(24,39,41,37,29,43,40,34);(2,4)(5,7,8,11)(6,9,10,12)(13,15,17,25,18,26,19,27)(14,20,33,42,23,36,22,35)(16,29,31,41,30,24,32,40)(21,37,44,39,38,34,28,43);(6,10)(7,11)(13,16,19,32,18,30,17,31)(14,21,33,44,23,38,22,28)(15,24,27,41,26,29,25,40)(20,34,42,43,36,37,35,39);(2,4)(16,30)(21,38)(24,29)(28,44)(31,32)(34,37)(39,43)(40,41)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^c=di

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, 277521);