#7049478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#31063	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fg,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313813, #72313897, #72314044
maximal normal subgroup of	#72313813, #72313897, #72314044

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6956674);