#6573698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62988	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6,11,16)(7,17,20,29)(8,13,22,28)(9,15,10,14)(12,23,26,19)(18,32,35,31)(21,30,24,27)(25,33,36,34);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,23)(11,20)(13,25)(14,17)(15,29)(16,26)(21,33)(22,35)(24,34)(27,31)(28,36)(30,32);(2,4)(5,8,11,22)(6,13,16,28)(7,18,20,35)(9,21,10,24)(12,25,26,36)(14,27,15,30)(17,31,29,32)(19,33,23,34);(1,2)(3,4)(5,9,11,10)(6,14,16,15)(7,19,20,23)(8,21,22,24)(12,17,26,29)(13,27,28,30)(18,33,35,34)(25,31,36,32);(2,4)(5,9,11,10)(6,14,16,15)(7,19,20,23)(8,21,22,24)(12,17,26,29)(13,27,28,30)(18,33,35,34)(25,31,36,32)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312395, #72312406, #72312412, …
maximal normal subgroup of	#72312395, #72312406, #72312412, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480894);