#154072

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

4218 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,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, 61268);