#356743

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

1550 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)(13,14,17,21,18,22,26,28)(15,19,16,24,23,27,25,20);(6,7)(9,10,11,12)(15,23)(16,25)(19,27)(20,24);(2,4)(13,15,17,16,18,23,26,25)(14,19,21,24,22,27,28,20);(5,7,8,6)(13,16,17,23,18,25,26,15)(14,20,21,19,22,24,28,27)

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,d^a=di,c^b=ci,d^b=dei,d^c=di

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, 263939);