#6413438

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2428	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fh,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343763, #72343884, #72343887
maximal normal subgroup of	#72343763, #72343884, #72343887

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6320634);