#359081

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249848, #72252972, #72256886
maximal normal subgroup of	#72249848, #72252972, #72256886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266277);