#359158

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252691, #72256869, #72256886, …
maximal normal subgroup of	#72252691, #72256869, #72256886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266354);