#2320704

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2216 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,20,16,17)(21,22)(23,28)(24,25)(26,27);(5,7)(6,10)(8,11)(9,12)(14,19)(15,16)(21,22,26,27)(23,28,24,25);(6,8)(10,11)(13,15,18,16)(14,20,19,17)(21,23,26,24)(22,25,27,28);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,15)(14,20,19,17)(21,24,26,23)(22,25,27,28);(5,8,9,6)(7,11,12,10)(13,17,18,20)(14,15,19,16)(21,25)(22,23)(24,27)(26,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=h,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2227900);