#423776

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1309 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330972);