#3335808

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

3572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,e^2=f,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	#72333451, #72333457, #72333494
maximal normal subgroup of	#72333451, #72333457, #72333494

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3243004);