#333286

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25631	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240482);