#503173

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3292	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1136 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fhi,d^2=ei,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=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, 410369);