#1127453

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2806 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)(3,7)(4,8)(5,6)(10,13)(14,15)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,32)(29,31);(2,6)(4,7)(9,10)(11,15)(12,13)(14,16)(17,19,20,24)(18,21,22,23)(25,27)(26,28)(29,32)(30,31);(2,6)(3,8)(10,13)(14,15)(18,22)(21,23)(25,27,29,32)(26,28,31,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232648, #72232649, #72232674
maximal normal subgroup of	#72232648, #72232649, #72232674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1034649);