#7514192

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286041, #72286108
maximal normal subgroup of	#72286041, #72286108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421388);