#3192274

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2778 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)(3,8)(4,5)(6,7)(9,10,13,15)(11,12,16,14)(17,18,21,24)(19,20,23,22)(25,26)(27,29)(28,30)(31,32);(1,3)(2,5)(4,6)(7,8)(9,11,13,16)(10,14,15,12)(17,19,21,23)(18,22,24,20)(25,26,28,30)(27,31,32,29);(2,7)(3,4)(10,15)(12,14)(19,23)(20,22)(25,26,28,30)(27,29,32,31);(1,4,6,3)(2,5,7,8)(9,12,13,14)(10,11,15,16)(17,20,21,22)(18,23,24,19)(25,26)(27,31)(28,30)(29,32);(1,5)(2,4)(3,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgi,b^2=gh,c^2=fhi,d^2=hi,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	#72334217, #72334228, #72334239
maximal normal subgroup of	#72334217, #72334228, #72334239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3099470);