#4973771

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fi,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304036, #72304043, #72304136
maximal normal subgroup of	#72304036, #72304043, #72304136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4880967);