#5021148

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

3364 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333437, #72333442, #72333494
maximal normal subgroup of	#72333437, #72333442, #72333494

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4928344);