#357612

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314370, #72314377, #72314395, …
maximal normal subgroup of	#72314370, #72314377, #72314395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264808);