#1492838

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂×#2668	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3244 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328184, #72328221, #72328296
maximal normal subgroup of	#72328184, #72328221, #72328296
maximal quotient of	#72330704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1400034);