#4101135

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,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, 4008331);