#354311

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214887, #72311842, #72311849, …
maximal normal subgroup of	#72214887, #72311842, #72311849, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261507);