#3395069

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3549 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,32)(28,29)(30,31);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,22,24,23)(26,31)(27,28);(3,7)(6,8)(11,15)(14,16)(18,20)(22,23)(25,27,30,28)(26,32,31,29);(1,2)(3,8)(4,5)(6,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,23)(21,24)(25,28,30,27)(26,29,31,32);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,22,24,23)(25,29)(26,28)(27,31)(30,32)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328661, #72328664, #72328723
maximal normal subgroup of	#72328661, #72328664, #72328723
maximal quotient of	#72331393

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3302265);