#7523582

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#35680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2561 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,24,28,31)(20,29,21,25)(26,27,30,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,26)(22,28)(23,31)(25,27)(29,32);(10,14)(11,15)(17,20,22,21)(18,25,23,29)(19,26,28,30)(24,32,31,27);(10,14)(13,16)(17,21,22,20)(18,25,23,29)(19,27,28,32)(24,30,31,26);(2,6)(5,7)(17,18,22,23)(19,24,28,31)(20,29,21,25)(26,27,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308707, #72308745
maximal normal subgroup of	#72308707, #72308745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430778);