#499948

order	512
nilpotency class	4
isoclinism	isoclinic class of #499946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407144);