#2844111

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3007 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232599, #72232604, #72232632
maximal normal subgroup of	#72232599, #72232604, #72232632
maximal quotient of	#72328534, #72328618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2751307);