#417733

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2373 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,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	#72255889, #72255908, #72255916
maximal normal subgroup of	#72255889, #72255908, #72255916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324929);