#514179

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch

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, 421375);