#6437942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62852	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4722 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:

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

Polycyclic presentation:

2^9|a^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285925, #72286018, #72286253, …
maximal normal subgroup of	#72285925, #72286018, #72286253, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345138);