#4422594

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

3648 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300480, #72300525, #72300539
maximal normal subgroup of	#72300480, #72300525, #72300539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4329790);