#356738

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

1546 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)(5,6,8,7)(13,14,17,21,18,22,23,26)(15,19,24,27,16,20,25,28);(6,7)(9,10,11,12)(15,16)(19,20)(24,25)(27,28);(2,4)(13,15,17,24,18,16,23,25)(14,19,21,27,22,20,26,28);(5,7,8,6)(13,16,17,25,18,15,23,24)(14,20,21,28,22,19,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263934);