#4068839

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2866 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326740, #72326749, #72326769
maximal normal subgroup of	#72326740, #72326749, #72326769

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3976035);