#6342383

order	512
nilpotency class	2
isoclinism	isoclinic class of #6342371
automorphism	Aut(G) order 100663296
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

8085 conjugacy classes of subgroups; only the 978 normal subgroups are shown.

normal subgroups only

This group has 8085 conjugacy classes of subgroups — too many to draw. Only its 978 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334924, #72334927
maximal normal subgroup of	#72334924, #72334927
maximal quotient of	#72334965

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249579);