#417661

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2483 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314248, #72314253, #72314335
maximal normal subgroup of	#72314248, #72314253, #72314335
maximal quotient of	#72314542, #72314562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324857);