#1397969

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3896 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)(6,9)(7,11)(14,17)(15,19)(22,25)(23,27);(5,6)(7,10)(8,9)(11,12)(21,22)(23,26)(24,25)(27,28);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15,16,19)(14,18,17,20)(21,22)(23,28)(24,25)(26,27);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328559, #72328573, #72328704, …
maximal normal subgroup of	#72328559, #72328573, #72328704, …
maximal quotient of	#72330685, #72332594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1305165);