#1254394

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,4)(3,6,7,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,24)(19,22,20,23)(25,26,29,30)(27,28,32,31);(2,4)(6,8)(9,10,13,11)(12,15,16,14)(17,19,21,20)(18,22,24,23)(26,30)(28,31);(3,7)(6,8)(10,11)(14,15)(25,27,29,32)(26,31,30,28);(1,3,5,7)(2,6,4,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31);(1,4,5,2)(3,8,7,6)(9,12,13,16)(10,14,11,15)(17,20,21,19)(18,23,24,22)(25,28)(26,27)(29,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300711, #72300750, #72300781, …
maximal normal subgroup of	#72300711, #72300750, #72300781, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1161590);