#1006125

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232340, #72232368, #72232419, …
maximal normal subgroup of	#72232340, #72232368, #72232419, …
maximal quotient of	#72323629, #72329916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 913321);