#3003825

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3428 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,19)(17,18)(21,22)(23,27)(24,25)(26,28);(2,4)(5,6,8,10)(7,9,12,11)(14,17)(15,16)(21,23,26,24)(22,25,28,27);(1,2)(3,4)(13,15,18,16)(14,19,17,20)(21,23,26,24)(22,27,28,25);(5,7,8,12)(6,9,10,11)(13,16,18,15)(14,19,17,20)(21,24,26,23)(22,27,28,25);(5,6)(7,11)(8,10)(9,12)(13,17,18,14)(15,20,16,19)(21,25,26,27)(22,24,28,23)

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,d^2=f,e^2=gi,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	#72297707, #72297708, #72297764
maximal normal subgroup of	#72297707, #72297708, #72297764
maximal quotient of	#72298348

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2911021);