#2131808

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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
prime radical	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3406 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)(6,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,27)(23,26,28,25);(5,6,8,9)(7,10,11,12)(13,14,17,16)(15,18,20,19)(21,23)(22,25)(24,28)(26,27);(7,11)(10,12)(14,16)(18,19)(21,23,24,28)(22,26,27,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,22)(23,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,23)(22,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fgh,c^2=gh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293940, C₂×#2131808
maximal normal subgroup of	#72293940, C₂×#2131808
maximal quotient of	#72293964, C₂×#2131808

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2039004);