#6442724

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4746 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:

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

Polycyclic presentation:

2^9|c^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282288, #72282332, #72283263, …
maximal normal subgroup of	#72282288, #72282332, #72283263, …
maximal quotient of	#72219648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349920);