#626969

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

2408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(9,10,14,11)(12,16,13,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(2,4)(5,6,7,8)(9,11,14,10)(12,16,13,15)(18,22)(19,26)(24,28)(27,32);(1,2,3,4)(5,6)(7,8)(9,12,14,13)(10,15,11,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(9,13,14,12)(10,16,11,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 534165);