#149433

order	512
nilpotency class	4
isoclinism	isoclinic class of #129414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56629);