#2125184

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2669	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3641 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)(3,4)(5,6,7,8)(10,13)(14,16)(17,18)(19,20)(21,23)(22,24);(1,2,4,3)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,23,20);(2,3)(11,15)(14,16)(19,24)(20,22);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,23)(22,24);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,22)(23,24)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,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	#72328292, #72328321, #72328333
maximal normal subgroup of	#72328292, #72328321, #72328333
maximal quotient of	#72330585

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2032380);