#517749

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424945);