#359035

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23406	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252628, #72256876, #72256886, …
maximal normal subgroup of	#72252628, #72256876, #72256886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266231);