#1809453

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3557 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336620, #72336637, #72336704
maximal normal subgroup of	#72336620, #72336637, #72336704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1716649);