#417663

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
prime radical	#60369	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2403 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:

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

Polycyclic presentation:

2^9|a^2=i,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	#72314237, #72314253, #72314334
maximal normal subgroup of	#72314237, #72314253, #72314334
maximal quotient of	#72314501, #72314520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324859);