#502096

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2192	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409292);