#789602

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

4222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263597, #72263606, #72263620
maximal normal subgroup of	#72263597, #72263606, #72263620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 696798);