#192090

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1296 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 266 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 — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99286);