#2601504

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
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

4253 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,19,20,17)(21,22,25,28)(23,27,24,26);(5,7,9,8)(6,10,11,12)(15,20)(17,19)(21,23,25,24)(22,26,28,27);(13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,27)(24,26)(25,28);(5,8,9,7)(6,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,23)(22,27,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336688, #72336732, #72336741
maximal normal subgroup of	#72336688, #72336732, #72336741

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2508700);