#6508701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3549	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3376 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279530, #72308580, #72308811
maximal normal subgroup of	#72279530, #72308580, #72308811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415897);