#759708

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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	#14757	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3224 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)(5,6,9,12)(7,11,8,10)(13,14,16,17)(15,19,20,18)(22,24)(26,28);(5,7,9,8)(6,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,26,27,28);(7,8)(10,11)(14,17)(18,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,16,20)(14,18,17,19);(5,8,9,7)(6,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24,25,22)(23,26,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=g,e^2=fg,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.

Relationships

maximal subgroup of	#72300470, #72300473, #72300570
maximal normal subgroup of	#72300470, #72300473, #72300570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 666904);