#6379032

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	C₂²×C₄	C₂⁵
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4585 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)(5,6,9,14,10,13,22,27)(7,11,17,25,16,24,32,35)(8,18,19,28,20,12,21,26)(15,29,31,36,30,23,33,34);(1,2)(3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,23)(13,24)(14,25)(18,29)(19,31)(20,30)(21,33)(22,32)(26,34)(27,35)(28,36);(5,8,10,20)(6,12,13,18)(7,15,16,30)(9,21,22,19)(11,23,24,29)(14,28,27,26)(17,33,32,31)(25,36,35,34);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,23)(13,24)(14,25)(18,29)(19,31)(20,30)(21,33)(22,32)(26,34)(27,35)(28,36);(7,16)(11,24)(15,30)(17,32)(23,29)(25,35)(31,33)(34,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,h^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	#72313645, #72313659, #72313693
maximal normal subgroup of	#72313645, #72313659, #72313693

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286228);