#4932981

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

2847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329612, #72329635, #72329718
maximal normal subgroup of	#72329612, #72329635, #72329718

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4840177);