#1000327

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2068 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)(6,11)(9,12)(13,14)(15,16)(17,19)(18,20);(5,6)(7,9)(8,12)(10,11)(13,15,17,20)(14,16,19,18)(21,22,24,26)(23,25,27,28);(5,7,10,8)(6,12,11,9)(13,14,17,19)(15,18,20,16)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,15)(14,18)(16,19)(17,20);(5,9,10,12)(6,7,11,8)(13,16,17,18)(14,15,19,20)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|b^2=fh,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 907523);