#362121

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 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)(5,6,10,15,11,9,26,14)(7,16,20,27,19,32,18,12)(8,21,23,30,24,13,25,31)(17,29,35,28,34,33,22,36);(5,7,10,20,11,19,26,18)(6,12,15,16,9,27,14,32)(8,22,25,34,24,35,23,17)(13,29,30,36,21,33,31,28);(1,2,3,4)(5,8,11,24)(6,13,9,21)(7,17,19,34)(10,23,26,25)(12,28,27,36)(14,30,15,31)(16,33,32,29)(18,22,20,35);(5,9)(6,11)(7,12)(8,21)(10,14)(13,24)(15,26)(16,20)(17,28)(18,32)(19,27)(22,29)(23,30)(25,31)(33,35)(34,36)

Polycyclic presentation:

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

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, 269317);