#861523

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3277 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334086, #72334106, #72334125
maximal normal subgroup of	#72334086, #72334106, #72334125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 768719);