#3297891

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
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	#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3006 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fgh,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334169, #72334202, #72334220
maximal normal subgroup of	#72334169, #72334202, #72334220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3205087);