#6371505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342612
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

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×#3485	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

6326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318776, #72319996, #72320194
maximal normal subgroup of	#72318776, #72319996, #72320194

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278701);