#4940623

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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

1784 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)(7,11)(8,10)(9,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,35)(31,32,37,36)(33,38,41,42)(39,40,44,43);(5,7,10,9)(6,11,8,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,37)(33,41)(40,43);(5,7,10,9)(6,12,8,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31,34,37)(30,36,35,32)(33,39,41,44)(38,43,42,40);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,36)(30,37,35,31)(33,40,41,43)(38,44,42,39);(5,9,10,7)(6,12,8,11)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33,34,41)(30,38,35,42)(31,39,37,44)(32,40,36,43)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=fgh,e^2=fgh,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, 4847819);