#7608648

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#23850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 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)(11,13)(12,14)(15,16)(19,26)(23,31)(27,32)(28,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,27,26,32)(20,24,21,30)(23,28,31,29);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(19,26)(23,31)(27,32)(28,29);(10,14)(13,16)(17,19,20,28,22,26,21,29)(18,23,30,32,25,31,24,27);(2,5)(6,8)(17,20,22,21)(18,24,25,30)(19,28,26,29)(23,27,31,32)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=hi,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,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257558, #72257577, #72264275
maximal normal subgroup of	#72257558, #72257577, #72264275
maximal quotient of	#72307902

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7515844);