#417564

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2432 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)(5,6,10,14)(7,11,17,26)(8,12,18,27)(9,20,21,13)(15,29,22,24)(16,32,34,25)(19,30,23,28)(31,35,33,36);(1,2)(3,4)(7,15)(9,21)(11,24)(13,20)(16,33)(17,22)(19,23)(25,36)(26,29)(28,30)(31,34)(32,35);(5,7)(6,11)(8,16)(9,22)(10,17)(12,25)(13,29)(14,26)(15,21)(18,34)(19,31)(20,24)(23,33)(27,32)(28,35)(30,36);(2,4)(5,8)(6,12)(7,16)(9,23)(10,18)(11,25)(13,30)(14,27)(15,31)(17,34)(19,21)(20,28)(22,33)(24,35)(26,32)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318110, #72318259, #72318305
maximal normal subgroup of	#72318110, #72318259, #72318305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324760);