#154026

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄³

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

3330 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eg,d^2=gh,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de

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, 61222);