#2036836

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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

2516 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)(15,17,16,20)(21,22)(23,25)(24,26)(27,28);(6,9)(7,12)(14,19)(17,20)(22,24)(25,27);(5,7)(6,10)(8,12)(9,11)(13,15,18,16)(14,17,19,20)(21,23,26,28)(22,27,24,25);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,18,15)(14,20,19,17)(21,24,26,22)(23,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,17,18,20)(14,15,19,16)(21,25)(22,23)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=fg,c^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1944032);