#1000324

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2227 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)(5,6,10,8)(7,12,11,9)(13,14,18,19)(15,17,16,20)(21,22)(23,24)(25,27)(26,28);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(21,23,25,28)(22,24,27,26);(7,11)(9,12)(13,15,18,16)(14,20,19,17)(21,22,25,27)(23,26,28,24);(1,2)(3,4)(5,8,10,6)(7,12,11,9)(13,16,18,15)(14,20,19,17)(21,23)(22,26)(24,27)(25,28);(5,9,10,12)(6,11,8,7)(13,17,18,20)(14,15,19,16)(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=g,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, 907520);