#921793

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3886 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(22,25)(23,27);(6,9)(10,11)(13,14,16,17)(15,18,20,19)(21,22)(23,26)(24,25)(27,28);(6,9)(10,11)(14,17)(18,19)(23,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15,16,20)(14,18,17,19)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256748, #72256757, #72256786, …
maximal normal subgroup of	#72256748, #72256757, #72256786, …
maximal quotient of	#72332951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 828989);