#7590124

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23861	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257340, #72257577, #72264023
maximal normal subgroup of	#72257340, #72257577, #72264023
maximal quotient of	#72307794

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497320);