#449685

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2617	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356881);