#357133

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1356 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,g^2=i,b^a=be,c^a=cf,c^b=ci,d^b=dei,d^c=di

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, 264329);