#545600

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 452796);