#362296

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

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, 269492);