#4936289

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,9,11)(7,10,8,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30)(31,32)(33,36)(34,35)(37,40)(38,39)(41,42)(43,44);(5,7,9,8)(6,10,11,12)(15,22)(19,21)(23,28)(26,27)(30,35)(31,37)(33,41)(39,44);(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28)(29,31,34,37)(30,32,35,40)(33,38,41,43)(36,39,42,44);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32)(30,31)(33,39)(34,40)(35,37)(36,38)(41,44)(42,43);(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,41)(30,36,35,42)(31,38,37,43)(32,39,40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=gh,d^2=fh,e^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4843485);