#6379035

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×#3499	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4564 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)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,23,29)(8,20,21,30,22,13,24,28)(17,31,33,35,34,25,32,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,23)(10,18)(11,19)(13,25)(14,29)(15,27)(16,26)(20,31)(21,33)(22,34)(24,32)(28,36)(30,35);(5,8,11,22)(6,13,15,20)(7,17,19,34)(9,21,10,24)(12,25,27,31)(14,28,16,30)(18,32,23,33)(26,35,29,36);(2,4)(5,7)(6,12)(8,17)(9,23)(10,18)(11,19)(13,25)(14,29)(15,27)(16,26)(20,31)(21,33)(22,34)(24,32)(28,36)(30,35);(5,9,11,10)(6,14,15,16)(7,18,19,23)(8,21,22,24)(12,26,27,29)(13,28,20,30)(17,32,34,33)(25,35,31,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^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	#72312395, #72312403, #72312814, …
maximal normal subgroup of	#72312395, #72312403, #72312814, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286231);