#4101132

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

1998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

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, 4008328);