#760285

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 667481);