#427295

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334491);