#7501009

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

3237 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)(5,6,9,15,11,17,10,16)(7,18,21,29,20,30,22,12)(8,13,24,31,25,32,26,14)(19,28,23,33,34,36,35,27);(5,7)(6,12)(8,23)(9,22)(10,21)(11,20)(13,28)(14,33)(15,30)(16,18)(17,29)(19,24)(25,35)(26,34)(27,31)(32,36);(2,4)(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,27)(14,16)(15,31)(17,32)(18,28)(20,34)(21,23)(22,35)(29,33)(30,36);(2,4)(5,8)(6,14)(7,19)(9,26)(10,24)(11,25)(12,28)(13,16)(15,32)(17,31)(18,27)(20,34)(21,35)(22,23)(29,36)(30,33);(7,20)(12,29)(18,30)(19,34)(21,22)(23,35)(27,33)(28,36)

Polycyclic presentation:

2^9|a^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250779, #72256516, #72256521
maximal normal subgroup of	#72250779, #72256516, #72256521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408205);