#1494876

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=gh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267449
maximal normal subgroup of	#72267449
maximal quotient of	#72267414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1402072);