#6370835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278031);