#154070

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2714 conjugacy classes of subgroups; only the 646 normal subgroups are shown.

normal subgroups only

This group has 2714 conjugacy classes of subgroups — too many to draw. Only its 646 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6,8,7)(10,13)(14,16)(25,26,28,29)(27,30,32,31);(1,2,4,3)(5,6,8,7)(10,13)(11,15)(18,21)(22,24)(26,29)(30,31);(9,10,12,13)(11,14,15,16)(17,18,20,21)(19,22,23,24)(25,26)(27,31)(28,29)(30,32);(1,3,4,2)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eg,d^2=fgi,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72238837
Fitting subgroup of	#72238837
maximal subgroup of	#72238837
maximal normal subgroup of	#72238837
maximal quotient of	#72238817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61266);