#2942462

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3478 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=gi,e^2=fg,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	#72326299, #72326331, #72326359
maximal normal subgroup of	#72326299, #72326331, #72326359
maximal quotient of	#72329097, #72332790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2849658);