#368415

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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)(7,10)(8,17)(9,23)(11,15)(12,16)(13,21)(14,24)(18,26)(19,27)(20,29)(22,28)(25,30)(31,32)(33,35)(34,36);(1,3,4,2)(5,7,11,16,12,15,10,6)(8,13,21,17,22,29,20,28)(9,18,25,30,26,23,24,14)(19,27,33,31,34,36,32,35);(5,8,11,21,12,22,10,20)(6,13,15,28,16,29,7,17)(9,19,25,33,26,34,24,32)(14,27,23,35,30,36,18,31);(5,9,11,25,12,26,10,24)(6,14,7,18,16,30,15,23)(8,19,21,33,22,34,20,32)(13,27,17,31,29,36,28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312212, #72312252, #72312300
maximal normal subgroup of	#72312212, #72312252, #72312300

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275611);