#2617246

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3358 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234030, #72234039, #72234041
maximal normal subgroup of	#72234030, #72234039, #72234041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2524442);