#192329

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1390 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309261, #72309265, #72309281, …
maximal normal subgroup of	#72309261, #72309265, #72309281, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99525);