#6379017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342728
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4628 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306236, #72306238, #72306269, …
maximal normal subgroup of	#72306236, #72306238, #72306269, …
maximal quotient of	#72306428, #72306435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286213);