#495074

order	512
nilpotency class	4
isoclinism	isoclinic class of #483425
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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)(5,6,8,14,11,16,10,26)(7,13,19,32,20,33,17,31)(9,22,21,34,25,27,24,15)(12,23,29,18,30,35,28,36);(5,7)(6,12,16,30)(8,17)(9,23)(10,19)(11,20)(13,27,33,22)(14,28,26,29)(15,32,34,31)(18,24)(21,36)(25,35);(1,2,3,4)(5,7)(6,13)(8,19)(9,18)(10,17)(11,20)(12,27)(14,32)(15,29)(16,33)(21,35)(22,30)(23,24)(25,36)(26,31)(28,34);(5,8,11,10)(6,14,16,26)(7,17,20,19)(9,21,25,24)(12,28,30,29)(13,31,33,32)(15,22,34,27)(18,23,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237237, #72239540, #72239554, …
maximal normal subgroup of	#72237237, #72239540, #72239554, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402270);