#7622824

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285780, #72320425, #72320537, …
maximal normal subgroup of	#72285780, #72320425, #72320537, …
maximal quotient of	#72285952, #72286953, #72287145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530020);