#533006

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#2464	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3108 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fgi,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef

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, 440202);