#496359

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cei,f^c=fh,h^c=hi

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, 403555);