#392144

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3994 conjugacy classes of subgroups; only the 586 normal subgroups are shown.

normal subgroups only

This group has 3994 conjugacy classes of subgroups — too many to draw. Only its 586 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286783, #72286798, #72286847
maximal normal subgroup of	#72286783, #72286798, #72286847
maximal quotient of	#72287818

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299340);