#376657

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(6,11)(8,20)(9,12)(13,22)(14,28)(15,24)(16,29)(17,31)(18,25)(21,26)(27,30)(32,35);(5,6,12,24,10,15,9,11)(7,16,18,29,19,13,25,22)(8,14,26,34,21,28,20,23)(17,30,31,36,32,27,35,33);(5,7,9,18,10,19,12,25)(6,13,11,22,15,16,24,29)(8,17,20,31,21,32,26,35)(14,27,23,33,28,30,34,36);(1,3,4,2)(5,8,12,26,10,21,9,20)(6,14,24,34,15,28,11,23)(7,17,25,35,19,32,18,31)(13,27,29,36,16,30,22,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283853);