#7531702

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	#45353	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,25)(23,26)(27,29)(28,31)(30,32);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,22,26,24)(21,25,30,32)(27,29,28,31);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,23,19)(18,22,26,24)(21,28,30,27)(25,31,32,29);(10,14)(13,15)(17,21,23,30)(18,25,26,32)(19,27,20,28)(22,31,24,29);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,19,26,20)(21,29,30,31)(25,28,32,27)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=i,e^2=i,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,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281600, #72282274
maximal normal subgroup of	#72281600, #72282274

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7438898);