#156342

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3146 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=efh,d^2=fg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63538);